M-estimators are a type of estimator in statistics that are defined by their robustness to outliers in the data. In other words, M-estimators are designed to produce accurate estimates even when the data contains a few unusually large or small values that might otherwise throw off the results.

One of the most commonly used M-estimators is the median absolute deviation (MAD), which is a measure of the dispersion of a dataset. To calculate the MAD, you first need to find the median of the data. This is the value that falls in the middle of the dataset when the values are ordered from smallest to largest. Then, for each value in the dataset, you calculate the absolute difference between that value and the median. The MAD is then defined as the median of these absolute differences.

The MAD is a useful measure of dispersion because it is relatively robust to outliers. For example, consider a dataset that contains 99 values that are all very close to each other, and one value that is much larger or smaller than the others. If we calculate the standard deviation of this dataset, the outlier value will have a large influence on the result, and the standard deviation will be much larger than it would be if the outlier value were not present. In contrast, the MAD will be relatively unaffected by the presence of the outlier, because the median of the absolute differences will not be greatly influenced by a single large or small value.

Another commonly used M-estimator is the Tukey biweight, which is a measure of the location of a dataset. To calculate the Tukey biweight, you first need to calculate the median of the data. Then, for each value in the dataset, you calculate the difference between that value and the median, and square the result. The Tukey biweight is then defined as the sum of the squares of these differences, divided by a certain constant that depends on the number of values in the dataset.

The Tukey biweight is a useful measure of location because it is relatively robust to outliers. For example, consider a dataset that contains 99 values that are all very close to the median, and one value that is much larger or smaller than the others. If we calculate the mean of this dataset, the outlier value will have a large influence on the result, and the mean will be much further from the median than it would be if the outlier value were not present. In contrast, the Tukey biweight will be relatively unaffected by the presence of the outlier, because the sum of the squares of the differences will not be greatly influenced by a single large or small value.